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STAT 111 Introductory Statistics. Lecture 7: More on Random Variables, Probability, and Sampling May 27, 2004. Today’s Topics. Finishing up mean and variance Conditional Probability Multiplication Rule and Independence Tree Diagrams Bayes’s Rule Sampling distributions

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Detail 111 Introductory Statistics Lecture 7: More on Random Variables, Probability, and Sampling May 27, 2004

Today\'s Topics Finishing up mean and difference Conditional Probability Multiplication Rule and Independence Tree Diagrams Bayes\' Rule Sampling conveyances Binomial circulation for test tallies

Recall: Rules for Means Let X and Y be (not really free) arbitrary factors and let an and b be constants. Manage 1 E( a + bX ) = a + b E(X) Rule 2 E( a X + b Y ) = an E(X) + b E(Y) If X and Y are free, then E(XY) = E(X) E(Y)

Recall: Rules for Variances Let X and Y be irregular factors, and let an and b again be constants. Run 1 Var(a + b X) = b 2 Var(X) Rule 2 If X and Y are autonomous, then Var(X ± Y) = Var(X) + Var(Y) Rule 3 If X and Y have connection ρ , then Var(X ± Y) = Var(X) + Var(Y) ± 2 ρ σ X σ Y

Example: Pick 3 Ticket Suppose you purchase a $1 Pick 3 ticket on each of two distinctive days. The settlements X and Y on the two tickets are free. Let X + Y be the aggregate result. Ascertain Expected estimation of the aggregate result Variance of the aggregate result Standard deviation of the aggregate result

Example: Heights of Women The tallness of young ladies somewhere around 18 and 24 in America is roughly ordinarily disseminated with mean µ = 64.5 and s.d. σ = 2.5. Two ladies are arbitrarily browsed this age aggregate. What are the mean and s.d. of the distinction in their statures? What is the likelihood that one is no less than 5" taller than the other? What is the IQR of statures in this age assemble?

Conditional Probability The likelihood of an occasion can change on the off chance that we know some other occasion has happened. The contingent likelihood of an occasion gives us the likelihood of one occasion under the condition that we know the result of another occasion. Let An and B be any two occasions to such an extent that P(B) > 0. The contingent likelihood of An accepting that B has as of now happened is composed P(A | B) :

Example: Rolling Dice Let A be the occasion that a 4 shows up on a solitary move of a reasonable 6-sided kick the bucket, and let B be the occasion that a significantly number shows up. Find P(A | B) and P(B | A) . Assume we include another (distinctive shaded so we can recognize the two) bite the dust to the blend, and let C be the occasion that the total of the two dice is more prominent than 8. Find P(A | C) and P(C | A) .

Example: Gender of Children Suppose we have a family with two youngsters. Expect every one of the four conceivable results ({older kid, more youthful boy},… ) are similarly likely. What is the likelihood that both are young ladies given that no less than one is a young lady? Assume rather that we overlooked the age of the kids and recognized just three family sorts. How might this change the above likelihood?

Example: Drawing Cards Draw 2 cards off the highest point of a very much rearranged deck. What is the likelihood that the second card is an Ace, given that the principal card was an Ace? Then again, consider just the primary card for a moment. Assume you don\'t see what the card is, and your companion lets you know the card is a King. What is the likelihood that the card is a precious stone?

Multiplication Rule The likelihood that both occasion An and occasion B happen is given by P(A and B) = P(A) P(B | A) = P(B) P(A | B) Here, P(A | B) and P(B | A) have the typical importance of being restrictive probabilities.

Example: Home Security House security specialists evaluate that an untrained house canine has a 70% likelihood of distinguishing a gatecrasher – and, given identification, a half shot of driving the interloper off. What is the likelihood that Fido effectively obstructs a thief? (The likelihood of a prepared guard dog distinguishing and running off an interloper is assessed to be around 0.75)

Example: Drawing Chips from a Urn A urn contains 5 white chips and 4 blue chips. Two chips are drawn successively and without substitution. What is the likelihood of getting the grouping (W, B)? The increase administer can be reached out to higher-arrange convergences. For instance, assume we toss 3 red chips and 5 yellow chips into our urn. Five chips are drawn successively and without substitution. What is the likelihood of getting the succession (W, R, W, B, Y)?

Independence Recall that two occasions An and B are autonomous if knowing one happens does not change the likelihood that alternate happens. At the point when two occasions are autonomous, we have that P(B | A) = P(B) and P(A | B) = P(A) Recall our case about the likelihood of a solitary card draw being a precious stone given that we are let it know is a King.

Many Independent Events Suppose we have n free occasions A 1 , A 2 , … , A n . At that point the duplication administer is

Example: Ten Rolls of a Die Roll a kick the bucket ten times. What is the likelihood that we roll a 2 10 times? What is the likelihood that we move no less than one 2?

Example: Height of Women Randomly select 8 American young ladies matured 18 to 24. What is the likelihood that each of the 8 ladies are more than 65 inches tall? What is the likelihood no less than one of the ladies is somewhere around 63 and 67 inches tall?

Tree Diagrams A tree graph is regularly useful for fathoming more intricate figurings, and specifically, issues that have a few phases. In a tree outline, every fragment in the tree speaks to one phase of the issue. Each entire branch demonstrates a conceivable way. Tree outlines consolidate both the expansion and increase rules.

Sample Tree Diagram Conditional likelihood of result in Stage 2 given the result in Stage 1 Stage 2 Probability of result in Stage 1 Stage 1 H 0.5 H Second flip 0.5 T First flip 0.5 H 0.5 Second flip T 0.5 T

HH HT TH TT Example: Tossing a Coin Twice P(HH) = P(H)P(H) = (0.5)(0.5) = 0.25 P(HT) = P(H)P(T) = (0.5)(0.5) = 0.25 P(TH) = P(T)P(H) = (0.5)(0.5) = 0.25 P(TT) = P(T)P(T) = (0.5)(0.5) = 0.25 H Second flip T First flip H Second flip T

Example: Dependent Coin Flips The past case has free coins flips, however we can envision circumstances where coin flips will be reliant. Consider the accompanying circumstance. We have two coins, one reasonable ( P(H) = 0.5 = P(T) ) and one-sided ( P(H) = 0.75 , P(T) = 0.25 ). Flip reasonable coin; if H, utilize one-sided coin next; something else, utilize reasonable coin once more. Flip one-sided coin after each H, reasonable coin after each T. Coin flips now are no more drawn out free.

Example: Chips in a Urn As a marginally unique case, consider a urn with 5 white chips and 4 blue chips. We draw three chips successively and without substitution. What is the likelihood of getting the succession (W, B, B)? Ascertain this utilizing a tree graph.

Example: Lab Testing A lab test can yield either a positive or negative result. For individuals with a specific sickness, it will create a positive result 90% of the time. Be that as it may, it will likewise deliver a constructive result in 0.1% of all solid individuals. Assume that 0.01% of the populace really has the sickness. What is the likelihood that an individual is solid? That a wiped out individual creates a negative test outcome? That an individual is sound and has a negative test outcome?

Ill and + Ill yet - Healthy however + Health and - Tree Diagram for Lab Testing Stage 2 Stage 1 + 0.90 sick 0.0001 Test ? - Individual 0.001 + ? solid Test ? -

More on Lab Testing We know from our underlying conditions that if a man picked haphazardly has the malady, we get a positive test with likelihood 0.90. What we\'re more inspired by more often than not is diagnosing people with the malady. At the end of the day, we need to comprehend what the likelihood is that an individual has the sickness given that his test outcome is sure.

Bayes\' Rule What we need is a strategy that permits us to utilize our known contingent probabilities to process the restrictive probabilities "in the other heading." The equation we utilize is called Bayes\' Rule and can be expressed as takes after: if An and B are any two occasions whose probabilities are not 0 or 1, then

Derivation of Bayes\' Rule

Example: Lab Testing In our illustration, An is the occasion that an individual is sick, and B is the occasion that the test outcome is sure. In this way, how about we figure the likelihood of an individual being sick given a positive test outcome.

Example: Coins and Urns A one-sided coin, twice as prone to come up heads as it is tails, is hurled once. In the event that heads, draw chip from urn I, which contains 3 white chips and 4 red chips. On the off chance that tails, draw chip from urn II, which contains 6 white chips and 3 red chips. Given that a white chip was drawn, what is the likelihood that the coin came up tails?

Population and Sampling Distributions The populace dissemination of a variable is the dispersion of its qualities for all individuals from the populace. The populace conveyance is additionally the likelihood circulation of the variable when we pick one individual from the populace at arbitrary.

Population and Sampling Distributions A measurement is any numeric measure that is utilized to portray the information we acquire. In the event that the information are gotten utilizing irregular testing, a measurement is an arbitrary variable, and henceforth its esteem changes from test to test. The likelihood conveyance of the measurement is known as its testing dissemination .

Population and Sampling Distributions The examining dispersion of a measurement depends on the populace circulation, as well as on the example estimate and the strategy used to gather the information from the populace. A measurement can be utilized to evaluate the parameter of the populace.

Example: Two Different Surveys Two national studies are wanted to gauge the extent p of peo